《American Mathematical Society Colloquium Publications Volume XX Interpolation and Approximation》

CHAPTER ⅠPOSSIBILITY OF APPROXIMATION;ANALYTIC FUNCTIONS1

1．1．Point sets:preliminary definitions1

1．2．Function-theoretic considerations4

1．3．An open set as the sum of regions6

1．4．Expansion of an analytic function10

1．5．A theorem on analytic extension12

1．6．Approximation;choice of poles14

1．7．Components of an analytic function17

1．8．Methods of Appell and of Wolff19

1．9．On the vanishing of analytic functions22

1．10．Necessary conditions for approximation23

CHAPTER ⅡPOSSIBILITY OF APPROXIMATION,CONTINOED27

2．1．Lindelof＇s first theorem27

2．2．Lindelof＇s second theorem29

2．3．Conformal mapping of variable regions32

2．4．Approximation in a closed Jordan region36

2．5．Applications,Jordan configurations39

2．6．General forms of Cauchy＇s integral formula42

2．7．Surface integrals as measures of approximation44

2．8．Uniform approximation;further results46

CHAPTER ⅢINTERPOLATION AND LEMNISCATES49

3．1．Polynomials of interpolation49

3．2．Sequences and series of interpolation52

3．3．Lemniscates and the Jacobi series54

3．4．An analogous series of interpolation56

3．5．A more general series of interpolation60

CHAPTER ⅣDEGREE OF CONVERGENCE OF POLYNOMIALS．OVERCONVERGENCE65

4．1．Equipotential curves in conformal maps65

4．2．Approximation of Jordan curves by a lemniscate68

4．3．Approximation of modulus of the mapping function71

4．4．Approximation of modulus of mapping function,continued74

4．5．Degree of convergence．Sufficient conditions75

4．6．Degree of convergence．Necessary conditions．Overconvergence77

4．7．Maximal convergence79

4．8．Exact regions of uniform convergence83

4．9．Approximation on more general point sets (irregular case)85

CHAPTER ⅤBEST APPROXIMATION BY POLYNOMIALS89

5．1．Tchebycheff approximation89

5．2．Approximation measured by a line integral91

5．3．Approximation measured by a surface integral95

5．4．Approximation measured by a line integral after conformal mapping of complement98

5．5．Approximation measured by a line integral after conformal mapping of interior100

5．6．Point sets with infinitely many components102

5．7．Generality of weight functions104

5．8．Approximation of functions not analytic on closed set considered107

CHAPTER ⅥORTHOGONALITY AND LEAST WQUARES111

6．1．Orthogonal functions and least squares111

6．2．Orthogonalization113

6．3．Riesz-Fischer theory116

6．4．Closure120

6．5．Polynomial approximation to analytic functions125

6．6．Asymptotic properties of coefficients128

6．7．Regions of convergence131

6．8．Polynomials orthogonal on several curves133

6．9．Functions of the second kind136

6．10．Functions of class H2141

6．11．Polynomials in z and 1/z143

6．12．An extremal problem,line integrals146

6．13．An extremal problem,surface integrals149

CHAPTER ⅦINTERPOLATION BY POLYNOMIALS152

7．1．Interpolation in roots of unity152

7．2．A sufficient condition for maximal convergence154

7．3．A necessary condition for uniform convergence159

7．4．Further conditions for maximal convergence162

7．5．Uniform distribution of points164

7．6．Interpolation in points uniformly distributed167

7．7．Points of interpolation with extremal properties170

7．8．Existence of polynomials converging maximally (Shen)173

7．9．A synthesis of interpolation and Tchebycheff approximation175

7．10．Least squares and interpolation in roots of unity178

CHAPTER ⅧINTERPOLATION BY RATIONAL FUNCTIONS184

8．1．Interpolation formulas184

8．2．Sequences and series of interpolation188

8．3．Duality:general theorems193

8．4．Duality:illustrations199

8．5．Duality and series of interpolation203

8．6．Illustrations206

8．7．Harmonic functions as generating functions209

8．8．Harmonic functions as generating functions,continued212

8．9．Geometric conditions on given points218

8．10．Geometric conditions,continuation221

CHAPTER ⅨAPPROXIMATION BY RATIONAL FUNCTIONS224

9．1．Least squares on the unit circle and interpolation224

9．2．Unit circle．Convergence theorems228

9．3．Unit circle．Other measures of approximation231

9．4．Unit circle．Asymptotic conditions on poles235

9．5．Applications239

9．6．Poles with limit points on circumference243

9．7．General point sets;degree of convergence249

9．8．General point sets;best approximation252

9．9．Extensions257

9．10．General point sets;asymptotic conditions on poles261

9．11．Operations with asymptotic conditions265

9．12．Asymptotic conditions under conformal transformation270

9．13．Further problems276

CHAPTER ⅩINTERPOLATION AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE281

10．1．The Blaschke product281

10．2．Functions of modulus not greater than M286

10．3．Functions of least maximum modulus290

10．4．Convergence of minimizing sequences293

10．5．Totality of interpolating functions296

10．6．Conditions for uniqueness300

10．7．Functions of class H2304

CHAPTER ⅪAPPROXIMATION WITH AUXILIARY CONDITIONS AND TO NON-ANALYTIC FUNCTIONS310

11．1．Approximation with interpolation to given function310

11．2．Interpolation to given function;degree of convergence314

11．3．Extremal problems involving auxiliary conditions318

11．4．Expansion of mapping function322

11．5．Approximation on a rectifiable Jordan curve328

11．6．Interpolation in roots of unity333

11．7．Tchebycheff measure of approximation;extremal problems335

11．8．Tchebycheff approximation by polynomials and rational functions338

11．9．Approximation by non-vanishing functions343

CHAPTER ⅫEXISTENCE AND UNIQUENESS OF RATIONAL FUNCTIONS OF BEST APPROXIMATION348

12．1．Sequences of rational functions of given degree348

12．2．Application to Tchebycheff approximation351

12．3．Restriction of location of poles353

12．4．The rational function of best approximation need not be unique356

12．5．Integral measures of approximation357

12．6．Approximation with auxiliary conditions360

12．7．Uniqueness of approximating functions with preassigned poles361

A1．Possibility of approximation of polynomials367

A2．Approximation by polynomials．Continuity conditions371

A3．Interpolation and approximation by bounded analytic functions373

BIBLIOGRAPHY379

INDEX391